Launcher opens, click Play, Launcher closes... Nothing else happens.

Go into the game folder, run the other executable... the game opens but in a window and is partially off screen.

Click New, Yes... Briefly see a dragon statue then it drops to my desktop. No error message or acknowledgment that it crashed.

Identical behavior with installer 1.6.659.0.8 and installer 0.1.3905696.

Identical behavior when Run as Administrator.

Which game? Are we playing guessing games?

Seeing " 1.6.659.0.8 ", I guess Skyrim.
In any case, everybody should always put the game name in the title or at the beginning of the post since this is a forum for the entire Elder Scrolls series.

I start on this page.
https://www.gog.com/en/game/the_elder_scrolls_v_skyrim_special_edition

I click on "Forum discussion".

I wind up here.

Why is this not the forum for The Elder Scrolls V: Skyrim Special Edition?

This is a forum for the entire Elder Scrolls series. As in ALL the games in the series share this forum.
GOG decided that there was not enough activity for each game to get its own forum, so they share this one.
So now you know.

That is a horrible approach and not at all what anyone would expect when seeking help with a specific game.

I agree (and thanks for explaining why you didn't mention Skyrim in the thread title - but the forum page says clearly "Elder Scrolls Series", and two of the first threads have "Morrowind" and "Oblivion" in their titles).

I'd like to help you, but I'm not familiar enough with Skyrim to be of much use.